8.311: Electromagnetic Theory
Problem Set # 11
Due: 4/28/04
Retarded fields, dipole radiation, antennas.
Reading: Schwinger, Chaps. 31, 32, 33, 34.
1. Field buildup around a straight wire.
Current in an infinite straight wire is turned on abruptly at t = 0:
�
I(t) =
0, t < 0
I0 , t > 0
The wire remains neutral at all times, ρ = 0.
a) Using the retarded potentials formula, find A(r, t) and Φ(r, t) around the wire.
b) From the potentials, determine the electric and magnetic field E(r, t), B(r, t). Plot the fields
schematically as a function of time.
c) Find the energy flux at a distance r from the wire, given by the Poynting vector S = 4cπ E × B
integrated over the surface of a cylinder of radius r and length L.
2. Dipole radiation near a surface.
Consider a radiating dipole placed near a surface of perfect conductor. The effect of the conductor
can be described by an image dipole, with the total radiation field being a sum of the contributions
due to the primary dipole and its image. Assume that the image dipole oscillates in phase with the
primary dipole, which is a reasonable approximation when the distance from the dipole to the surface
is smaller than the radiation wavelength λ = 2π/k = 2πc/ω with ω the dipole oscillation frequency.
For a dipole oscillating at an angle θ with respect to the normal, d(t) = d0 cos ωt, find the radiated
power. At which values of θ the radiated power is enhanced, and for which θ it is suppressed due
to the proximity to the conductor? (This problem provides a simple explanation of the so­called
Peierls effect, describing the change in luminescence of atoms and molecules adsorbed on a surface
of a metal.)
3. Antenna radiation. (Schwinger, Problem 6, Chap 34)
Obtain formulas for the angular distribution of radiated power for a half­wave center­fed antenna,
kl = π, and for a full wave antenna. Plot the latter, and compare with Figure 34.4.